#4928710

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,7)(10,13)(11,15)(19,23)(22,24)(25,26,30,31)(27,32,34,28)(29,37,39,33)(35,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,22,23,24)(26,31)(27,34)(33,37)(35,40);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=fghi,d^2=gh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4835906);